And as if single-phase circuits aren’t already complicated enough to analyze, three-phase circuits show up. So far in my tutorials for this subject, we’ve been dealing with single-phase circuits. A **single-phase** AC power system consists of a generator (or the source) connected through a pair of wires (a transmission line) to a load (or the impedance). So if you were to duplicate a single-phase circuit twice, you would have a three-phase circuit. As easy as that. Not really.

A **three-phase system** is produced by a generator consisting of three sources having the same amplitude (or voltages) and frequency but their phase angles have a difference of 120º. It is considered as by far the most prevalent and most economical polyphase* system, due to the amount of wire required for a three-phase system being less than that required for an equivalent single-phase system. When one-phase or two-phase inputs are required, they are taken from the three-phase system rather than generated independently. That’s why nearly all electric power is generated and distributed in three-phase.

**circuits or systems in which the AC sources operate at the same frequency but different phases.*

Here’s how a (a) two-wire type and a (b) three-wire type look like. Note that they are both single-phase systems.

Just because it’s single-phase, doesn’t mean it only has a single loop. A single-phase system means the phase angles are equal, I think. While there’s a single-phase and a three-phase, a two-phase system is also possible.

As you can see, the generator or sources of the two-phase have equal amplitudes, yet having a different value for the phase angle. While the phase angles of a three-phase have a difference of 120º, two-phase have a difference of only 90º.

Here’s a design of a three-phase system:

When a three-phase circuit possess the phase angle difference of its voltage sources but still having its magnitudes equal, it is considered as a *balanced three-phase circuit* or to contain *balanced phase voltages*.

A three-phase with balanced phase voltages will look like this in a graph, or whatever this is that displays sinusoids and stuff:

And here is how it is illustrated on a phasor diagram, with the phase sequences: (a) *abc* or positive sequence, and (b) *acb* or negative sequence:

So, what about this *abc* and *acb* sequence? Well, since the three-phase voltages are 120º out of phase with each other, there are two possible combinations for this. One is the (a) *abc* or positive sequence phasor diagram where *Van* leads *Vbn*, which in turn leads *Vcn*. This possibility is mathematically expressed as:

Van = Vp∠0º

Vbn = Vp∠-120º

Vcn = Vp∠+120º

*Vp* is the **effective** or **rms value** of the phase voltage. The other possibility is the (b) *acb* or negative sequence phasor diagram where *Van* leads *Vcn*, which in turn leads *Vbn*. This possibility is given by:

Van = Vp∠0º

Vbn = Vp∠+120º

Vcn = Vp∠-120º

You must have at least wondered how these three balanced voltage sources are connected to each other in a three-phase system, right? No? Me too. There are actually two types of how the three sources (and the three loads) are connected to each other. And those two are called: the **wye**-connection **Y**, and the **delta**-connection **Δ**.

The one on the left are (a) Y-connected sources, while the other on the right are (b) Δ-connected sources. The three-phase system design that I showed above is an example of a wye-wye connection.

To convert Y-connected sources to Δ-connected sources, or vice versa, you go:

Vl = √3 · Vp ∠(θ + 30º)

Vp = Vl/√3 ∠(θ – 30º)

Where:

Vp = |Van| = |Vbn| = |Vcn|

Vl = |Vab| = |Vbc| = |Vca|

It is a common tradition in power systems that the voltage and current, when it comes to three-phase circuits, are always in **rms** values, unless stated otherwise.

The sources in a three-phase circuit are not the only ones connected in wye or delta, but also the loads.

A set of (a) Y-connected loads and a set of (b) Δ-connected loads. And of course, there is a *balanced load*, which is considered when the phase impedances are equal in magnitude and in phase. A Y-connected load consists of three impedances connected to a neutral node, while a Δ-connected load consists of three impedances connected around a loop. The load is balanced when the three impedances are equal in either case.

For a balanced Y-connected load,

Z1 = Z2 = Z3 = Zy

Where *Zy* is the load impedance per phase. While for a balanced Δ-connected load,

Za = Zb = Zc = ZΔ

So, to convert a Y-connected load to a Δ-connected load, or vice versa, you go:

ZΔ = 3Zy

Zy = ZΔ/3

To solve for the current *I*, just convert the three-phase circuit to a balanced wye-wye connection then get its single-phase equivalent:

Ia = Van/Zy

Ib = Vbn/Zy

Ic = Vcn/Zy

If it is for a balanced delta-delta connection, the formulas would just go:

Ia = Iab – Ica

Ib = Ibc – Iab

Ic = Ica – Ibc

Where:

Iab = Vab/ZΔ

Ibc = Vbc/ZΔ

Ica = Vca/ZΔ

And that’s all there is for three-phase circuits. If I have missed some informations or you have some questions about the topic, just comment it down below.